Small oscillations: diagonal matrix

1. The problem statement, all variables and given/known data
I'm solving an exercise about small oscillations.

I name [itex]T[/itex] the kinetic matrix and $H$ the hessian matrix of potential.

The matrix [itex]\omega^2 T- H[/itex] is diagonal and so find the auto-frequencies is easy! But I have a problem with normal modes. The lagrangian coordinates are two angles, [itex]\theta[/itex] and [itex]\phi[/itex].

Normal modes are given by splitted oscillations of the two coordinates. Is it correct? Are they given by:
[itex]\theta(t)=A_1 \cos(\Omega t+ \alpha_1)[/itex] and [itex]\phi(t)=A_2(\cos \frac{k}{M} t+\alpha_2)[/itex]? ([itex]A_1, A_2[/itex]= constants depending on initial conditions)

And is the general solution of motion given by $$\theta(t)+\phi(t)=A_1 \cos(\Omega t+ \alpha_1)+A_2(\cos \frac{k}{M} t+\alpha_2)$$?

yeah, that all agrees with the matrix equation you wrote in the first post. Except one thing. You have written the eigenvectors the wrong way around. The whole idea of the normal mode method is that the matrix you wrote, when acting on an eigenvector, should give a zero vector. So when omega squared equal k/M, then what should the eigenvector be?

Edit: woops, ah sorry sorry. You did write them the correct way around. I didn't take the time to look at the matrix product carefully enough. So all your work is correct.